GBM random walk – again

Mostly this write-up will cover the discrete-time process. In continuous, it’s no longer a walk [1]. Binomial tree, Monte Carlo and matlab are discrete.

Let’s divide the total timespan T — from last-observed to Expiry — into n equal intervals. At each step, look at ln(S_new/S_old), denoted r. (Notation is important in this field. It’s extremely useful to develop ascii-friendly symbols…) It’s good to denote the current step as Step “i”, so first step has i=1 i.e. r_1=ln(S_1/S_0). Let’s denote interval length as h=T/n.

To keep things simple let’s ignore the up/down and talk about the step size only. Here’s the key point —

Each step size such as our r_i is ~norm(0, h). r_i is non-deterministic, as if controlled by a computer. If we generate 1000 “realizations” of this one-step stoch process, we get 1000 r_i values. We would see a bell-shaped histogram.

What’s the “h” in the norm()? Well, this bell has a stdev, whose value depends on h. Given this is a Wiener process, sigma = sqrt(h). In other words, at each step the change is an independent random sample from a normal bell “generator” whose stdev = sqrt(step interval)

[1] more like a victim of incessant disturbance/jolt/bombardment. The magnitude of each movement would be smaller if the observation interval shortens so the path is continuous (– an invariant result independent of which realization we pick). However, the same path isn’t smooth or differentiable. On the surface, if we take one particular “realization” with interval=1microsec, we see many knee joints, but still a section (a sub-interval) may appear smooth. However, that’s the end-to-end aggregate movement over that interval. Zooming into one such smooth-looking section of the path, now with a new interval=1nanosec, we are likely to see knees, virtually guaranteed given the Wiener definition. If not in every interval then in most intervals. If not in this realization then in other realizations. Note a knee joint is not always zigzag . If 2 consecutive intervals see identical increments then the path is smooth, otherwise the 2-interval section may look like a reversal or a broken stick.

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